Reduced space charge models

[cemitch99]

Reduced space charge models, which make simplifying assumptions about the detailed form of the bunch distribution function and return space charge forces based on the 2nd beam moments (envelopes), are important for the modeling of circular machines. Such a model is also needed for envelope tracking with space charge (in lieu of the rms envelope equations).

Option 1: Nonlinear 2D/3D space charge model

• Input: bunch 2nd moments involving spatial coordinates & initial particle phase space 6-vector
• Output: final particle phase space 6-vector after applying a kick due to nonlinear space charge forces
• Assumption: the bunch charge distribution is described by a multivariate Gaussian with the appropriate moments

One could also use the linearized space charge forces near the origin, which yields a result distinct from Option 2.

Option 2: Linear 2D/3D space charge model

• Input: bunch 2nd moments involving spatial coordinates
• Output: matrix describing a linear kick due to space charge forces
• Assumption: the bunch charge distribution is described by a uniform ellipsoid with the appropriate moments

To be used in a follow-up to PR #814